1. Agenda Brief overview of dilute ultra-cold gases
Quantum dynamics of atomic matter waves
The Bose Hubbard and Hubbard models

2. Quantum degenerate Bose gas
High temperature T: Thermal velocity v Density d “billiard balls”
Low temperature T: De Broglie wavelength lDB=h/mv~T-1/ “Wave packets”
T=Tcrit : Bose Einstein Condensation De Broglie wavelength lDB=d “Matter wave overlap”
T=0 : Pure Bose Einstein Condensate
“Giant matter wave ”
Ketterle

3. BEC in dilute ultracold gases
In 1995 (70 years after Einstein’s prediction) teams in Colorado and Massachusetts achieved BEC in super-cold gas.This feat earned those scientists the 2001 Nobel Prize in physics.
S. Bose, 1924
A. Einstein, 1925
E. Cornell
C. Wieman
W. Ketterle
Light
Atoms
Using Rb and Na atoms

5. How about Fermions?
At T<Tf ~Tc fermions form a degenerate Fermi gas
𝐸 𝐹 ~ ℏ 2 2𝑚 3 𝜋 2 𝑛 2/3
1999: 40 K JILA, Debbie Jin group

6. To cool the atoms down we need to apply lasers in all three directions
Laser cooling
An atom with velocity V is illuminated with a laser with appropriate frequency
Trapping and cooling: MOT
Atom
To cool the atoms down we need to apply lasers in all three directions
laser
Atoms absorb light and reduce their speed
Slower atom
Laser cooling is not enough to cool down the atoms to quantum degeneracy and other tecniques are required
As the atoms slow down the gas is cooled down

7. Evaporative cooling Temperatures down to 10-100 nanoK
Each atom behaves as a bar magnet
This process is similar to what happens with your cup of coffee.
The hottest molecules escape from the cup as vapor
In a magnetic field atoms can be trapped: magnetic field~coffee cup
By changing the magnetic field the hot atoms escape and only the ones that are cold enough remain trapped
2001
BEC

8. How to see a BEC ? T>Tc T<Tc T=Tc Time of flight images
t=0 Turn off trapping potentials
Light Probe
Image
T>Tc
T<Tc
T=Tc
MPI,Munich

9. Anisotropic

10. Characterizes low energy collisions
Scattering in quantum mechanics
(1) Particles behave like waves (T → 0)
− ℏ 2 2𝜇 𝛻 2 Ψ+𝑉(𝑟) Ψ=EΨ
There is only a phase shift at long range!!
Phase shift at low energy proportional to
“scattering length” 𝑎
Characterizes low energy collisions

11. Basic Scattering in quantum mechanics
Goal: find scattered wave
f: scattering amplitude
∝ 𝑒 𝑖𝑘𝑧 +𝑓(Ω) 𝑒 𝑖𝑘𝑟 𝑟
(r) = ℓ 𝑅 ℓ (𝑟,𝐸)𝑃 ℓ ( cos 𝜃) l = 0, 1, 2… s-, p-, waves, …
There is only a phase shift at long range!!
𝑅 ℓ (𝑟→∞)→ 1 𝑘𝑟 sin 𝑘𝑟−ℓ 𝜋 2 − 𝛿 ℓ

12. Basic quantum scattering: s-wave
cross section
atoms/s scattered flux into d
plane wave flux (atoms/cm2/s)
Solve Schrödinger equation for each l
𝛿 ℓ
Quantum statistics matter
Pauli Exclusion principle
Get phase shift
𝛿 ℓ
Identical bosons: even
Identical fermions: odd
Non-identical species: all
𝛿 ℓ 𝑘 → 𝐴 ℓ ( 𝐴 ℓ 𝑘) 2ℓ
𝑘→0
𝐴 0 =a
“scattering length”
Characterize s-wave collisions
𝐴 1 =b3
“scattering volume ”
Characterize p-wave collisions

14. Pseudo-Potential
• Two interaction potentials V and V’ are equivalent if they have the same scattering length
• So: after measuring a for the real system, we can model with a very simple potential.
• Actually, to avoid divergences you need
Huang and Yang, Phys. Rev. 105, 767 (1957)
Also E. Fermi (1936), Breit (1947),
Blatt and Weisskopf (1952)

15. Weakly interacting gases
Quantum phenomena on a macroscopic scale.
Ultra cold gases are dilute
a: Scattering Length
n: Density
Cold gases have almost 100% condensate fraction In contrast to other superfluids like liquid Helium which have at most 10%

16. Many body Bosonic systems
Field operator
Many body Hamiltonian
Equation of Motion
In general
Ψ ′ =0

17. Gross Pitaevskii Equation
Dilute ultracold bosonic atoms are easy to model
1. Short range interactions
2. At T=0 all share the same macroscopic wave function
Ψ Φ =Φ Φ
Gross-Pitaevskii Equation
We can understand the many-body system by a single non linear equation

18. BEC and quantum coherence
A BEC is a coherent collection of atomic deBroglie waves
laser light is a coherent collection of electromagnetic waves
laser

19. Quantum coherence at macroscopic level
Vortices
JILA 2002
MIT (1997).
Coherence
Weakly interacting Bose Gas
Superflow
Non Linear optics
MIT (1997).
NIST (1999).